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Physica C 468 (2008) 730–732

A study of superconducting transition of network modelsof multiply connected superconductors

Osamu Sato a,*, Masaru Kato b,c

a Osaka Prefectural College of Technology, Neyagawa 572-8572, Japanb Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan

c CREST-JST, Kawaguchi 332-0012, Japan

Accepted 30 November 2007Available online 29 February 2008

Abstract

We present external magnetic field dependence of superconducting transition temperature of multiply connected superconductors.The systems were regarded as superconducting wire networks and treated by the de Gennes–Alexander theory. We found that transitiontemperatures show various periodicities and stable states respect to field strength and field angle in the three-dimensional superconduc-ting networks.� 2008 Elsevier B.V. All rights reserved.

PACS: 74.20.De; 74.81.Fa

Keywords: Superconducting network; de Gennes–Alexander theory; Little-Parks oscillation

1. Introduction

Connectivity of the superconductor causes peculiar mag-netic responses because of quantization of the magneticflux. Nano-technologies have been realized various artificialshapes and connectivities of superconductors, particularlythe multiply connected superconductors composed of verythin wires is called superconducting networks. Well-con-trolled micro porous superconducting metals may becomethree-dimensional superconducting networks.

In this paper, we report the external magnetic fielddependence of phase transition temperature of supercon-ducting networks. In nano-sized networks, symmetries ofvortex arrangements are strongly affected by the symmetryof the systems [1–3]. In these systems, existence of anti-vor-tex is derived in theoretical approaches. Shapes of the net-works are also strongly affected the vortex arrangement.We treat superconducting networks that composed of

0921-4534/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.physc.2007.11.037

* Corresponding author. Tel.: +81 72 821 6401; fax: +81 72 821 0134.E-mail address: gpsato@las.osaka-pct.ac.jp (O. Sato).

superconducting straight wires. Each wire has a length ofa, and whose diameter is supposed to be small comparedwith the Ginzburg–Landau coherence length. In the fol-lowing, we present the magnetic field dependence of thetransition temperature of square lattice networks and cubiclattice networks with randomly lack of wires. This model isone of the random network models.

2. Formalism

We consider the square lattice network whose nodalpoints are expressed in the Cartesian coordinates as (x, y,0) = (anx, any, 0) {nx, and ny are integers}, and the cubic lat-tice network whose nodal points are (x, y, z) = (anx, any,anz) {nx, ny and nz are integers}. Applied field H in a direc-tion of (h, u) and its vector potential A are expressed as

H ¼ Hðsin h cos /; sin h sin /; cos hÞ; ð1Þ

A ¼ H2ðz sin h sin /� y cos h; x cos h� z sin h cos /;

y sin h cos /� x sin h sin /Þ; ð2Þ

Fig. 2. Falls of the transition temperatures of 20 � 20 square latticenetworks in magnetic field along z-axis. Rates of lack of wires are (A)

O. Sato, M. Kato / Physica C 468 (2008) 730–732 731

where h and / are angles of the spherical polar coordinates.In this report, the field angle u is supposed to be u = 0.

We denote the order parameter at nodal point i as Di.The de Gennes–Alexander equation [4,5] at nodal point i

is given as

1

ni

Xj

Dj expðici;jÞ ¼ Di cosan; ð3Þ

where the sum of j is taken over the nodes connected bywires with node i, ni is the number of wires connecting withnode i. The phase ci,j is given by the line integral along awire from i to j as

ci;j ¼2pU0

Z j

iA � ds: ð4Þ

Here U0 ¼ hc=2e is the flux quantum for superconductivity.The Little-Parks oscillation of a single square superconduc-ting loop that has an area of a2 shows a field periodicity ofH ¼ U0=a2. Then it is convenient to represent the magneticfield as the ‘filling field’ f that is defined by f ¼ Ha2=U0.

r = 0%, (B) r = 10%, and (C) r = 20%.

3. Result

3.1. Square lattice networks

A schematic illustration of the system is shown in Fig. 1.Lack of superconducting wire causes magnetic response ofthe network because different flux quantization paths arise.

Fig. 2 presents magnetic field dependence of suppressionof transition temperature. Here we calculated 20 � 20square network and that with random lack of wire, andthe magnetic field is applied in z-direction (perpendicularto the network). The curve A presents transition tempera-ture of square network. This curve shows a periodicity off = 1. There is a dip structure at the field of f = 1/2 wherevortices form a stable arrangement. The curves B and Cshow square networks with lack of wire. Rates of lackare r = 10% and 20%, respectively. These curves also showthe same periodicity and dip structure as the curve A. Inthe networks of randomly lack of wires, falls of the transi-tion temperatures are suppressed. Relaxation of constraint

Fig. 1. A schematic illustration of a square lattice network with randomlylack of wires.

of vortices arrangement occurs because of lost of symmetryof the network.

In the field of h direction, the transition temperaturecurve simply become that of the periodicity of f = 1/cosh.

3.2. Cubic lattice networks

In the Fig. 3, the curve A presents the transition temper-ature of the cubic lattice (6 � 6 � 6) in a magnetic field of z-direction. The curves B1 and B2 present networks of r =10%. Similarly, C1 and C2 present network of r = 20%. Incomparison with square networks, falls of transition tem-peratures are small. This is because superconducting wires

Fig. 3. Falls of the transition temperatures of 6 � 6 � 6 cubic latticenetworks in magnetic field along z-axis. Rates of lack of wires are (A)r = 0%, (B1) (B2) r = 10%, and (C1) (C2) r = 20%.

Fig. 4. Falls of the transition temperatures of 6 � 6 � 6 cubic latticenetworks in magnetic field of h = p/6 direction. Rates of lack of wires are(A) r = 0%, (B1) (B2) r = 10%, and (C1) (C2) r = 20%.

732 O. Sato, M. Kato / Physica C 468 (2008) 730–732

along the field gain the superconducting order. Obviously,in the magnetic field in z-direction, transition temperaturecurves have a periodicity of f = 1. Sample dependence ofthe transition temperature is rather large in high r samplebecause loops of various areas are appears in high rsamples.

Fig. 4 shows the transition temperatures in a magneticfield in the direction of h = p/6. The transition tempera-tures have no ‘perfect periodicity’ respect to the field. Eachcurve shows that there are local stable states in periodicityof f ¼ 2=

ffiffiffi3p

. This is because projection area of a2 to h = p/6 direction is

ffiffiffi3p

a2=2. In the field of h = p/4 direction, tran-

sition temperature of the networks have a perfect periodic-ity of f ¼

ffiffiffi2p

. This is because each loop in the cubic latticenetwork has a projection area can be expressed asa2=

ffiffiffi2p� n (n is an integer) in spite of lack of superconduc-

ting wires.In field of angle h 6¼ 0; p=4, the transition temperature

has no periodicity because ratios of projection areas ofloops to the h direction are irrational numbers.

4. Summary

We studied superconducting square and cubic latticenetworks with randomly lack of superconducting wires.In the networks of randomly lack of wires, falls of the tran-sition temperatures are suppressed. In the cubic networks,transition temperature shows various periodicities and sta-ble states respect to field strength and field angle.

Acknowledgement

This work was supported by Grant-in-Aid for ScientificResearch from the Ministry of Education, Culture, Sports,Science and Technology, and the Computing system ofInstitute for Materials Research (IMR), Tohoku University.

References

[1] O. Sato, M. Kato, Phys. Rev. B 68 (2003) 094509.[2] O. Sato, S. Takamori, M. Kato, Phys. Rev. B 69 (2004) 092505.[3] V.R. Misko, V.M. Fomin, J.T. Devreese, V.V. Moshchalkov, Phys.

Rev. Lett. 90 (2003) 147003.[4] P.G. de Gennes, C.R. Acad. Sci. Paris Ser. II 292 (1981) 279.[5] S. Alexander, Phys. Rev. B 27 (1983) 1541.